Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2204.10526

A survey on the parameterized complexity of the independent set and (connected) dominating set reconfiguration problems

Nicolas Bousquet·Amer E. Mouawad·Naomi Nishimura·Sebastian Siebertz
View original source ↗Read source document ↗Export metadata ↓

Abstract

A graph vertex-subset problem defines which subsets of the vertices of an input graph are feasible solutions. We view a feasible solution as a set of tokens placed on the vertices of the graph. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions of size $k$, whether it is possible to transform one into the other by a sequence of token slides (along edges of the graph) or token jumps (between arbitrary vertices of the graph) such that each intermediate set remains a feasible solution of size $k$. Many algorithmic questions present themselves in the form of reconfiguration problems: Given the description of an initial system state and the description of a target state, is it possible to transform the system from its initial state into the target one while preserving certain properties of the system in the process? Such questions have received a substantial amount of attention under the so-called combinatorial reconfiguration framework. We consider reconfiguration variants of three fundamental underlying graph vertex-subset problems, namely Independent Set, Dominating Set, and Connected Dominating Set. We survey both older and more recent work on the parameterized complexity of all three problems when parameterized by the number of tokens $k$. The emphasis will be on positive results and the most common techniques for the design of fixed-parameter tractable algorithms.

Explore related subjects

cs.CCcs.DMcs.DSmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nicolas Bousquet, Amer E. Mouawad, Naomi Nishimura, Sebastian Siebertz. 2022-04-22. A survey on the parameterized complexity of the independent set and (connected) dominating set reconfiguration problems. https://arxiv.org/abs/2204.10526

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Counting Small Induced Subgraphs: Hardness of Symmetry-Based Properties

Jerrum and Meeks (TOCT, JCSS 2015) introduced the counting problems $\text{IndSub}(Φ)$ for fixed graph properties $Φ$: Given an input graph $G$ and $k\in\mathbb N$, count the $k$-vertex subsets $S \subseteq V(G)$ such that the induced subgraph $G[S]$ satisfies $Φ$. For recursively enumerable $Φ$, it is known that $\text{IndSub}(Φ)$ is either #W[1]-hard or fixed-parameter tractable. A direct classification depending on $Φ$ however still remains open. In particular, the status was open for the property of graphs without nontrivial automorphisms, also mentioned in a very recent survey on parameterized counting by Roth (Comput.~Sci.~Rev.~2026). This is a natural property that evades all currently known techniques for proving #W[1]-hardness, including a general toolkit based on Fourier analysis that was very recently introduced by Curticapean and Neuen (SODA~2025). In this paper, we show that counting induced $k$-vertex graphs without nontrivial automorphisms is #W[1]-hard by constructing ``clique scaffolds'', i.e., problem-specific restrictions of the property that enable a reduction from the $k$-clique problem. More generally, we show that for every finite group $Q$, counting $k$-vertex induced subgraphs with automorphism group $Q$ is #W[1]-hard.

cs.CC↗
arXiv2026

Improved Algorithms for the Remote Point Problem

The Remote Point Problem (RPP) is an algorithmic problem that asks, given a linear subspace $L \subseteq \mathbb{F}^n$ of dimension $k$, to deterministically find a vector $v \in \mathbb{F}^n$ far in Hamming distance from $L$. This problem was introduced by Alon, Panigrahy and Yekhanin [APY09], motivated in part by the matrix rigidity approach for proving circuit lower bounds. An algorithm is said to achieve remoteness $d$ if it finds a vector $v$ whose Hamming distance from $L$ is at least $d$. We observe that over the rational numbers, the problem admits a deterministic polynomial-time algorithm that achieves optimal remoteness $n-k$. Over finite fields, we obtain a (modest) improvement of a result of Alon, Panigrahy and Yekhanin [APY09], and give an algorithm that achieves remoteness $Ω\left(\frac{n}{\max\{k, \log n\}} \log n\right)$.

cs.CC↗
arXiv2026

Strong Selective and List-Decoding Direct Product Theorems for Quantum Query Complexity

Quantum strong direct-product theorems for specific functions have been known for nearly two decades. These have been extended to general results for function computation and state generation. The proofs of these results use a version of the multiplicative adversary method that does not naturally extend to relations. Standard strong direct-product theorems apply when algorithms must correctly answer every given question. Prior work extended them to equivalent threshold direct-product theorems, which require answers to all questions but only require that most answers are correct. We focus on two further generalizations. Strong selective direct-products apply to algorithms that adaptively choose, based on what they learn from queries, which questions from a large list to answer. This generalization is relational and useful for proving time-space tradeoffs. We prove a quantum strong selective direct-product theorem for all functions using a new multiplicative adversary formulation for relations that satisfies a strong selective direct product property while being strong enough to capture any query lower bound for functions proven by negative-weights adversaries. This was not previously known even without selectivity. The second generalization is list-decoding direct product problems introduced by Ben-David and Blais for classical randomized query complexity. These allow an algorithm to produce a large list of possible output vectors such that one of them is fully correct. They proved that such theorems hold for classical randomized complexity of all Boolean functions. We prove a quantum analogue of this theorem for all partial Boolean functions. We show that strong list-decoding direct-product theorems are implied by a special case of multiplicative adversaries which we show, via a new reduction, can be obtained from negative-weights adversaries for any Boolean-valued function.

cs.CC↗